Problem: What is the extraneous solution to these equations? $\dfrac{x^2 - 47}{x + 10} = \dfrac{53}{x + 10}$
Explanation: Multiply both sides by $x + 10$ $ \dfrac{x^2 - 47}{x + 10} (x + 10) = \dfrac{53}{x + 10} (x + 10)$ $ x^2 - 47 = 53$ Subtract $53$ from both sides: $ x^2 - 47 - (53) = 53 - (53)$ $ x^2 - 47 - 53 = 0$ $ x^2 - 100 = 0$ Factor the expression: $ (x - 10)(x + 10) = 0$ Therefore $x = 10$ or $x = -10$ At $x = -10$ , the denominator of the original expression is 0. Since the expression is undefined at $x = -10$, it is an extraneous solution.